Keep getting zeros as Fourier Coefficients

Question

Ive got to solve the Laplace equation in polar form for $r>a$ (where $a$ is a constant) and I've boiled down the problem to $$ T=A_0+\sum^\infty_{n=1}\frac{B_n}{r^n}\cos(n\theta)+\frac{C_n}{r^n}\sin(n\theta) $$ by using the Boundary Value : $$ \lim_{r\to\infty} r\frac{\partial T} {\partial r} (r,\theta)=0. $$ I now want to use the Boundary Value $T(a,\theta)=T^*\cos^2(\theta)$ where $T^*$ is a constant, but I keep getting both fourier coefficients equal to zero. NB: I have got $A_0=\frac{T^*}{2}$ Any hints?

Answer 1

The quick and dirty way: $$ T^* \cos^2{\theta} = \frac{T^*}{2}(1+\cos{2\theta}). $$ Therefore by orthogonality, the only nonzero coefficients are of $1$ and $\cos{2\theta}$. Fiddling about with the constants gives $$ T = \frac{T^*}{2}\left(1+\frac{a^2}{r^2}\cos{2\theta}\right). $$

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Slightly less slick way: $\sin{n\theta}$ is odd, so $$ \int_{-\pi}^{\pi} \sin{n\theta}\cos^2{\theta}\, d\theta = 0, $$ and we can forget about the sine terms.

On the other hand, $$ \int_{-\pi}^{\pi} \cos{m\theta}\cos{n\theta} \, d\theta = \frac{1}{\pi}\delta_{mn} $$ for at least one of $n,m>0$, so the only term you have to worry about is the $\cos{2\theta}$ one, by $ 2\cos^{2}{\theta} = 1+ \cos{2\theta}$ as before. You might want to integrate it directly to check you know what you're doing, but otherwise, it's just a matter of getting the constants right, now that you know where to look.